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homework20111107 - Give the Laurent series expansion about...

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Math 417, Fall 2011 Exercise Set #7 Turn in: November 7 1. Find the maximum and minimum of | f ( z ) | on the unit disk { z C | | z | 1 } , where f ( z ) = z 2 2. 2. What is the radius of convergence for the Taylor series of the function f ( z ) = 1 z 2 3 z + 2 about z = 0? About z = 3 i ? 3. This problem has three parts: a) Find the power series representation for e az centered at z = 0, where a C is any constant b) Show that e z cos z = 1 2 e (1+ i ) z + e (1 i ) z . c) Find the power series expansion for f ( z ) = e z cos z centered at z = 0. 4. By integrating the power series expansion for f ( z ) = 1 1 + z 2 term by term, find a power series expansion for arctan z . What is the radius of convergence? 5. By di ff erentiating the geometric series for 1 1 z , find the Taylor series expansion of the following functions around z = 0: a ) 1 (1 z ) 2 b ) 1 (1 z ) 3 c ) 1 (1 z ) k 6. Give the
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Unformatted text preview: Give the Laurent series expansion about z = − 1 for the function f ( z ) = z 2 1 + z . 7. Find the Laurent series expansion of f ( z ) = 1 z 3 cos( 1 z 2 ) valid in the region | z | > 0. 8. Find the Frst three terms of the Laurent series for f ( z ) = 1 e z − 1 in the region < | z | < 2 π using long-division with the Taylor series of e z − 1. 9. Find the Laurent series expansion in terms of z of f ( z ) = z 1 + z in the region 1 < | z | < ∞ . Write your answer in summation form. 10. Find the Laurent series expansion in terms of z of the function f ( z ) = 1 z ( z 2 + 1) about z = 0 and about z = ∞ . 1...
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